If it's anything it will be laser thrusters, but that comes with needing major revolutions in the weight required to generate the necessary huge amounts of energy.
Erm. Relatively pretty strongly disagrees with you, otherwise c wouldn't be a constant.
Within the limits of non-speculative technology, you can't actually get constant acceleration from a propulsion source with a fixed impulse (all of them). There are diminishing returns the faster you travel. With rockets as an example, if you keep accelerating eventually you'll reach the point where your propellant out the back ends up a dead stop, after which you can't go any faster no matter how much fuel you have. Plus there's the issue of drag, which goes up with the square of the speed. The force isn't too high in the interstellar medium, even near c, but ship material integrity becomes a problem, and god help any organic life inside because at that speed you've converted the entire universe into ionizing radiation.