When moving to the US on a H1B or L1 visa, how are savings and current accounts in Europe taxed? Are the interest on them taxed? What about changes of exchange rate between USD/GBP/EUR from one year to the next?
If you're resident for tax purposes (which you are when you're on H1B/L1, except for may be the first/last years) they're taxed exactly as if they were in the US. You might not get 1099 from the bank (though some foreign banks do issue 1099 to US persons), but that is irrelevant to your tax obligations.
The interest is taxed in the year received. Currency fluctuations do not affect the interest taxation. You calculate the interest in USD based on the exchange rate in the day the interest was paid, and pay taxes based on the USD value.
Don't forget your FBARs (form 114) and FATCA forms, and don't forget to check the checkbox on Schedule B (line 7).
Pay attention: not reporting foreign accounts, especially if there's income (like interest income) may cost you dearly. $10000 per year per account for failing to file form 114, for example.