No state is listed on my 1099-DIV. I've got less than $2000 in exempt-interest dividends. Is that a small enough amount to allow me to not specify states?

If you did not have any state taxes withheld directly from the invested company, return to the "Tell us if any of these uncommon situations apply to you" screen and ensure that the "State taxes were withheld..." check box is clear (not checked). 

Then proceed to the next prompted question. If box 13 is blank on your 1099-DIV, it should mean that no taxes were withheld, and your 1099-DIV should not have any entries in boxes 13-15.

Enter the information from your 1099-DIV and the federal information will flow through to your state. 

the only reason that TurboTax asks for the state(s) is that some states exempt some types of their tax-exempt interest (which is reported as dividend income when paid by a mutual fund or ETF)  from taxation.    so it's a question of finding out from the fund the exact source and amount of those dividends that would be tax-exempt. if you can't get that info then it's taxable by your state if your state taxes such income  I find that even when you can get the info of the source and amount then it's up to you the taxpayer to find out if your state exempts it, many times you'll find the amount exempt is so small that it wasn't worth the effort.  I just use TX as the state.