Oh well, I did some research, and actually still do … here are the results I got so far.:
The first account dates from November 30th, 1895 and is given by a Dr. Georg Irmer, who was the Imperial German Government District Administrator in the Marshall Islands, which were a German overseas colony back then.
In his account he gives a bit information of some birds he saw when he inspected the Taongi Atoll (now Bokok) to collect guano samples for analysis and to reaffirm the German claim to the island, he mentiones seabirds and a large ground-dwelling bird which he named a ‘Trappe‘, the German term for a bustard. He gives no further description or whatsoever, but it is thought that he might not have seen any of the birds commonly known from the Marshall Islands because neither he nor his Marshallese crew were able to identify that bird.
Given his name for the bird, ‘Trappe‘, it is quite likely that he indeed saw a rail of the genus Gallirallus, very much like the one that once inhabited the Wake Atoll to the north of the Marshall Islands. 
The second account comes from the natives of the Marshall Islands and was forwarded by them to the German ‘anthropologists’ who explored these islands at the beginning of the 20th century.
It is a bird named as the anang-, annan-, or annang. This is said to have been a very small bird (the size of a butterfly (!)), and to have possessed a pleasant smell, it is said to have lived among the rocks around the shores of the northern Marshall Islands. The bird is known from oral traditions at least from the Jaluit-, and the Wotho Atoll, and it is always said to have been a ground-dwelling singing bird.
This may in fact be a description of a Turnstone (Arenaria interpres (L.)), a species that winters in Micronesia and that was very much appreciated, for example by the inhabitants of Nauru, who cought them not to eat them but to tame them and keep them as pets.
Or it is the description of a small crake or a reed-warbler, mixed with some phantastic components. 
The third account comes from Paul Hambruch, a German ethnologist that researched the life of the natives of the island of Nauru, his accounts are merely stories that were told him by a native named Auuiyeda, and which he translated into German.
Let’s read them.:
“Es gibt auch Vögel auf Nauru, wie Fregattvogel, schwarze Seeschwalbe, weiße Seeschwalbe, Regenpfeifer, Brachvogel, Möve, Schnepfe, Uferläufer, Ralle, Lachmöve und Rohrdrossel.” 
“There are also birds on Nauru, as frigate bird, black tern, white tern, plover, curlew, gull, snipe, sandpiper, rail, black-headed gull and reed thrush.“
And he goes on.:
“Die Vogelwelt ist nach Zahl und Art reicher. Der Fregattvogel (Tachypetes aquila), itsi, die schwarze Seeschwalbe (Anous), doror, die weiße Seeschwalbe (Gygis), dagiagia, werden als Haustiere gehalten; der erste galt früher als heiliger Vogel, mit den beiden anderen werden Kampfspiele veranstaltet. Am Strande trifft man den Steinwälzer (Strepsilas interpres), dagiduba, den Regenpfeifer (Numenius), den Uferläufer (Tringoides), ibibito, die Schnepfe, ikirer, den Brachvogel ikiuoi, den Strandreiter iuji, die Ralle, earero bauo und zwei Möwenarten (Sterna), igogora und ederakui.
Im Busche beobachtet man an den Blüten der Kokospalme den kleinen Honigsauger raigide, die Rohrdrossel (Calamoherpe syrinx), itirir und den Fliegenschnäpper (Rhipidura), temarubi.” 
“The bird world is richer by number and species, The frigate bird (Tachypetes aquila), itsi, the black tern (Anous), doror, the white tern (Gygis), dagiagia, are kept as pets; the first one was formerly considered a holy bird, with the two others are used for fighting games. At the beach one mets with the turnstone (Strepsilas interpres), dagiduba, the plover (Numenius), the sandpiper (Tringoides), ibibito, the snipe, ikirer, the curlew, ikiuoi, the beach rider [?] iuji, the rail, earero bauo and two gull species (Sterna), igogora and ederakui.
In the bush one observes on the flowers of the coconut palm the small honeyeater raigide, the reed thrush (Calamoherpe syrinx), itirir and the flycatcher (Rhipidura), temarubi.“
The author is usually thought to have misinterpreted the things he was told by Auuiyeda, but I personally doubt that somehow, all the mentioned landbirds make in fact sence for georaphical reasons, so, why not?
Nauru is now almost deserted, the whole island looks like a building site – and it actually is one! There are some sad rests of the forest that once covered the whole island, and indeed some landbirds still manage to survive in small numbers, one of them, the Nauru Reed-Warbler (Acrocephalus rehsei (Finsch)) is even an endemic species, there’s no reason not to accept the former presense of a fantail, a honeyeater, and especially a rail, no reason at all!
These birds, especially the rail, may already have been extirpated by the beginning of the 20th century, leaving only memories and storys told by the islanders.
And last but not least, here the fourth account of birds from the Namoluk Atoll, Chuuk, that were enumerated by Max Girschner, another German who had lived in Micronesia at the beginning of the 20th century, he was a colonial offical, a doctor, and a ethnologist.
I have no access to his accounts, but I can give you quotations of them by Mac Marshall from 1971, here they are.:
“Ponape Lory (no Namoluk name)
According to Girschner (1912:126), this species was blown to Namoluk in a typhoon in 1905, and apparently it still occurred on the atoll at the time of his visit. there are no lories at present on Namoluk nor can anyone alive on the atoll in 1971 remember seeing them.” 
According to Donald W. Buden this whole information is unlikely, and if these parrots have ever occurred on the Namoluk Atoll at all, they must have been brought there by people. 
I personally think … why not, typhoons may indeed blow parrots from one island to another, or how did the loris themselves came to end up on the island of Pohnpei in the first place?
But wait, there’s more.:
“A second bird mentioned by Girschner that no longer is found on Namoluk is “a small black and white bird” for which he gives the name lipukepuk.” 
The author states that this can only be the description of a New Hanover Mannikin (Lonchura (hunsteini ssp.) nigerrima (Rothschild & E. J. O. Hartert)), which does not occur anywhere in Micronesia and which is not black and white by the way. The bird he is actually referring to is Hunstein’s Mannikin (Lonchura hunsteini ssp. minor (Yamashina)), which again is very well occuring in Micronesia, at least on the island of Pohnpei (yes, again), and which is at least blackish and greyish ….
To me the whole account sounds very much like a nice description of the Truk Monarch (Monarcha rugensis (Hombron & Jacquinot)), and given the fact that most island-dwelling birds in Micronesia also occur on nearby atolls it is quite possible that there once was a native population of this bird here as well.
But we will probably never know for sure.
The most interesting things that I found out so far are.:
1: Micronesian bird names are odd (to my ears and eyes), I mean the Palau Ground Dove (Alopecoenas canifrons (Hartlaub & Finsch)) for example is named omekrengukl, I do not even know how to pronounce that. 🙂
2: Micronesia harbors only 148 native breeding bird species (including the extinct ones!).
3: The Micronesian landbirds do not only occur on the higher islands but also on the atolls, even on those atolls that are quite far away from the next high islands, a situation that is completely different from Polynesia, where the high islands almost entirely harbor a different avifauna than the atolls.
There may have been more species once, especially when we fill some of the illogical gaps between the islands and island groups.
 Paul Hambruch: Nauru. Ergebnisse der Südsee-Expedition 1908-1910. II. Ethnographie: B. Mikronesien, band 1.1 Halbband. Hamburg, Friedrichsen 1914
 Mac Marshall: The natural history of namoluk Atoll, eastern Caroline Islands. Atoll Research Bulletin 189: 1-53. 1975
 Donald W. Buden: The birds of Satawan Atoll and the Mortlock Islands, Chuuk, including the first record of Tree Martin Hirundo nigricans in Micronesia. Bulletin on the British Ornithologists’ Club 126(2): 137-152. 2006
 Dirk H. R. Spennemann: Extinctions and extirpations in Marshall Islands avifauna since European contact – a review of historic evidence. Micronesia 38(2): 253-266. 2006