I've been doing this since I got here.
In the US, I still have my credit union account, and a savings account with ING. Every month, I transfer money (with xe.com) to one/both of those accounts to pay bills/add to savings. (You do lose a little money, but at least the exchange rate is in your favour here!)
I keep a buffer in my credit union of about $500, just in case something comes up (I had to pay an annual storm/sewer fee for my house in NC, for example). Otherwise I just budget it into the amount I transfer over each month.
The transfer with xe takes 2-3 days, so I just do it the week before the bill is due.
Hope that helps!