If gravitation is due to space-time curvature, how can a body free-fall in a straight line? According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion?
The body must follow a curved path. So, there is no possibility of straight-line motion. In a curved space-time, there is no such thing as a straight line. If so, then how can there be a straight-line free fall?
 A: There are two answers to this. The simplest is that the curvature is small if you're far from any masses, so motion will be approximately in a straight line at a constant velocity.
The second answer is far more important, but also far harder to explain. Basically it's that we define a straight line as the trajectory followed by a freely moving particle. So for example a thrown object actually follows a straight line - it just looks like a parabola to us.
This may seem like playing with words, but actually if the geometry is non-Euclidean then there is no simple definition of a straight line. We are only used to having an intuitive grasp of what a straight line is because the geometry around the Earth's surface is approximately Euclidean. In a non-Euclidean geometry the principle we use is that if no force is acting on an object then that object will travel in a straight line. So a freely falling object travels in a straight line because, well, freely falling means no force is acting on it. The catch is that the straightness of a line is observer dependant, so different observers will disagree about whether a line is straight or not.
Jan's comment to your question refers to a geodesic, and this is the term for a straight line line in a non-Euclidean geometry. You may be interested to Google for geodesic to learn more, but to get any further than the simplified description I've given above will involve you getting stuck into the maths.
A: user36790, please don't take this answer the wrong way. It is not meant to be disparaging. Per your user page, you are 17 years old. You have some misunderstandings. You're way ahead of your peers, many of whom will hold similar (or even stronger) misunderstandings throughout their lives. You have asked a number of related questions over the last few hours. They all result from the same misunderstanding. This misunderstanding is that you are looking at things from a Newtonian point of view, where space is Euclidean, where time is an independent parameter, and where everyone agrees one what space and time are.
That's not how things work. It is very close to how things work under some special circumstances. Those special circumstances in which space and time locally appear to be distinct and Newtonian -- that's what we ordinarily experience on an everyday basis. This is why Newtonian mechanics has been so successful. That Newtonian mechanics works so well in our ordinary, everyday world does not mean that it is universally correct. In fact, we know it's not universally correct.

Then all paths must be curved. If so, how can there be any straight line motion?

This is your Newtonian mindset at work. Both special relativity and general relativity are markedly non-Euclidean. The sharp distinction between space and time in Newtonian mechanics becomes blurred in relativity theory; space and time become different aspects of one thing, spacetime.
Even though geometry in relativity theory is not Euclidean, one can still ask from the perspective of the non-Euclidean geometry of general relativity, "What is straight?" One definition of "straightness" in Euclidean geometry is that a straight line between two points is the path that has the shortest length amongst all the paths that connect the two points in question.
This concept of "straightness" extends nicely into the geometry of general relativity. All we need is something to measure "distance," a "metric," and this is something that general relativity provides. This generalization of a Euclidean straight line to a non-Euclidean geometry is called a "geodesic." 
A: You know what, you are absolutely correct. Nothing accelerates under the influence of gravity in straight lines. Rather, all objects follow the straightest and shortest paths available in that region of curved spacetime, called geodesics.
What is more confusing, however, is our perception and definition of a straight line. The concept of "straight" is based on our perception of the universe. To make the point clear, consider the following analogy:
Imagine a 2-dimensional creature living on a sheet of paper. It has no way of perceiving anything outside the sheet of paper. If you draw a line on the paper, the bug will see a straight line. Now imagine gently rolling the sheet of paper so that it is curved. The bug, who has no way of knowing that the sheet it lives on is curved, and can percieve things on the sheet only, will continue to see a straight line. We humans however, looking from above, can see the line is curved, simply because the paper it is drawn on is curved.
Something similar happens when objects accelerate under the influence of gravity. The path they follow is actually curved, but we percieve it as straight. 
Note that the anolgy is not entirely accurate. Curvature of a space, or spacetime, is an intrinsic property, and does not have anything to do with the way it is embedded in a higher dimension. Which means that you'll have to do more than roll a sheet of paper to actor introduce curvature whose effects can be felt within the sheet. Choosing a sphere and considering it a curved 2-D surface would be more accurate.
A: The simple answer is that if an object falls and accelerates, its 
X-T graph will be curved so its worldline would be curved so movement of the object will be a curve through space time and we only see space so we see it as a straight line.
But we can see curvature of spacetime in Earth revolving around the Sun 
