Is it safe for a company not to use any encryption on customer's data when it is transferred between data centers on dedicated lines?
By customer data I don't mean passowrds/credit card info. Just simply hosted customer data (files).
It is safe not to encrypt sensitive data in transit over some wires only insofar as no evildoer may physically access these wires at any point. So, in fact, no, it is not safe. In urban environments, cables run most of the time through tunnels shared with a lot of other usages (e.g. sewers or subway tunnels) and not under robust shielding or 24/7 guardianship. In the countryside, such links will often be buried in trenches not far from roads, and can be preyed upon by any attacker with some dedication and a shovel.
A 75 years old grandmother once shut down the quasi-totality of Internet access in Armenia in similar circumstances. Assuming that things are more "protected" in richer countries (like, say, Western Europe or the USA) would be a gross miscalculation of what cost/benefit analysis means. Cables are left unprotected because nobody would accept the price of physically protected cables, both in Armenia and in New Jersey.
Of course, that's only customer data. Nothing important, then.