Q:

Can you accelerate to almost the speed of light on a smooth, curved path?

So, many of the time dilation thought experiments have periods of accelerating away from the origin, stopping, turning around, and returning, which precipitates an observable time differential between the traveler and those remaining behind.

Is there a requirement that the path be straight line?
Is is possible to accelerate to almost the speed of light (or a large fraction thereof) along a curved path (say a circle or ellipse) rather than "out and back"?

I'm happy to put aside the realities of the movement of the universe (the planet won't be in the same place to return to later on) for this thought experiment insofar as it simplifies the question.
My intuition says it must be straight, but I cannot articulate why, my physics is rusty.

A:

If one wishes to nitpick (and I do), any acceleration is a curved path in four-dimensional space-time. If there is curvature in an object's path in three-dimensional space, that means there is a component of acceleration perpendicular to the object's path.
It certainly is possible to accelerate perpendicular to one's path, but the curvature that results is proportional the acceleration divided by the square of velocity. This means that for an object traveling close to light speed, the acceleration need for even a modest amount of curvature would be massive. 
How massive? Well, the event horizon of a black hole consists of the surface such that if a light ray were traveling along it, it would be kept on that surface by the force of gravity. So, basically, if you have an object that's going close to light speed, then to keep it in a circle, you need a force comparable to a that of a black hole whose event horizon is the same size as that circle.

