First, the bomber is probably not a right-winger because he tried to limit the effect of the bombing to property damage by using a PA system to broadcast warnings to evacuate. Right-wing terrorists generally do not care about human life. Consider the Oklahoma City bombing or the 9/11 attacks as two examples.
In fact, I was originally going to speculate that it was likely a left-wing group behind the bombings, specifically because of the warnings. For example, both the Provisional IRA and the Weather Underground Organization specialized in calling in warnings to evacuate areas where they had placed bombs.
Except now we are learning that it was a lone wolf attack. Most left-wing bombings have groups behind them, so I now believe, by process of elimination, the bombing is be motivated by personal grudges. The target seems to have been the AT&T building in Nashville.
AT&T is a big, bureaucratic business with many customers. It is not hard to imagine that the bomber was harboring a grudge against AT&T for some past business dealing that went wrong, or some act of corporate malfeasance.
That is just one possible reason; the motive could easily be a moonbat conspiracy theory, or a grudge against some other business in the same immediate area. Mental illness is also a possibility, given that it was evidently a suicide bombing.
But it does not appear at this stage likely to be politically-motivated terrorism.