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Any micro with a timer interrupt should work fine. On a PIC for example you rotate the current byte, the bit rotated out goes into the Carry flag and you send the Carry flag out to the Port Pin you're using as your serial out. Do that 8 times for the byte, move onto the next byte and so on.
If your sending large amount of data then a micro with a 16 kbyte pointers (HC's AVR's, 8052's and others) are nice but not necessary.