What do we know about currencies and interest rates?

Not enough.  Here are a few simple relationships, noting in advance that theory and empirics sometimes conflict.

1. A high nominal interest on a currency "ought to" predict forthcoming depreciation of that currency, on average.  After all, market equilibrium has to offer you more to hold this asset precisely because investors expect it to fall in value.  This is known as uncovered interest parity. 

In other words, exchange rates should not be a random walk.  But returns on currencies should be a random walk.  This means that future exchange rate movements should be predicted by the nominal interest rate differential, but additional exchange rate movements, adjusting for that differential, should behave like a random walk.   

2. In reality these relationships are not obviously true.  Many investors believe that the market overestimates the risk of high nominal interest rate currencies.  If you simply slotted your money into high nominal interest rate currencies from, say, 1970 to 1990, you would have done very very well.  This view, however, is not as popular as it once was.  Starting with the Asian financial crises of the 1990s, investors have been burned speculating in high nominal interest rate currencies.

3. In a world of multiple currencies, nominal interest rates in fact serve as the true real rates of return.  Let's say you live in America but hold Japanese yen.  (Are there always enough non-liquidity-constrained investors in this position and able to drive a market equilibrium?)  Your real rate of return in dollars depends on your nominal rate of return in yen plus your ability to convert yen into dollars.  Once you take the yen-dollar conversion rate into account, the rate of inflation in Japan should not matter.

4. Given #3, theory suggests that the real interest rate differential between Japanese yen and U.S. dollars should not, ceteris paribus (a daunting stipulation in this context), affect future exchange rates.  The nominal rates should be doing the work.  That being said, theory seems to be wrong.  Real interest rate differentials do have predictive power for future currency movements.

So the puzzle is this.  When it comes to future currency values, nominal interest rates do not obviously matter as much as they ought to, given theory.  Real interest rate differentials matter more than they ought to.  Also see my earlier post on microeconomic vs. macroeconomic perspectives on currency movements.