I visited Iceland about 6 month ago and became hooked, however there's a few areas I didn't get a chance to do and I'm converting a 20" foldy into a BMX style hybrid to go back and tackle them. I'm looking at around Mar/April for two/three weeks before the snow has melted as the run-offs were a particular challenge in June.
The main areas of interest are the F821 up and the F26 through the middle and back. It's likely the tracks will be shut to traffic and that should indicate what you'll be up against. It'll be a very rewarding challenge, but you will need suitable camping gear, knobbly tyres and a sense of humour!