Didn't seem to pose much of a problem for the first near century of the US. There wasn't much of the way in a semaphore system before electrical telegraphy took off in the 1840s, and yet, the US was able to function as a republic fairly well with nearly its current borders, before the telegraph lines truly tied the nation together.
Ironically, of course, right before it tore itself apart; but in such a fashion that never threatened to undermine the basic Republican system.
Yes, you had native americans to worry about but not quite the same as the various tribes moving in from the east like the Huns or Goths.
It's actually you who is conflating modern ideas of what a republic is ("Anything that isn't a monarchy") with classical concepts of republicanism. Right up until the 20th century the word 'republic' carried far richer connotations than just 'anything without a crown'.
http://plato.stanford.edu/entries/republicanism/