The best way to understand curved space-time is Einstein's way in 1907. Imagine all of space is filled with clocks which are held in place, but they need tick at different rates in order to stay simultaneous with each other. Near a massive object, the clocks tick slowly, away from masses, they tick faster.

Particles travel through space so that they locally take the path of maximum time between fixed endpoints, so that between endpoints which are close to a massive object, their path curves out a little, meaning that they are bent toward the massive object.

This is a statement of the Einstein 1907 theory of gravity, which he knew then would be the weak field, slow velocity approximation to Genera Relativity. It is counterintuitive for a few reasons:

• In geometry, straight line paths are minimum distance. In relativity the path is a local maximum. This is a consequence of the minus sign in the Pythagorean theorem in relativity. In relativity, unlike in geometry, the sum of the length of two legs of a triangle (when these are not imaginary) is always less than the third, so that straight lines maximize proper time.
• There is only one function which describes the curving of space time, and this is the clock rate. The curvature is determined by this clock rate, but it is purely a time curvature. Space is not curved at all.
• The geodesic motion is not trivial to see from the clock-rate description. You might naively think that to maximize the proper time you need to move away from massive objects, because time ticks slower near them. But the maximization is holding the endpoints fixed. To give an equation of motion without the concept of maximum proper time, you can just say that objects feel a force of attraction towards regions of slower clock-tick, and leave it at that. But this doesn't look like a geometrical condition (although it is).

I don't believe that there are two pictures of a phenomenon, one appropriate for laymen and a separate one for physicists. A correct picture is a correct picture, and is useful for both, and a misleading picture is misleading for both. This picture is used by all General Relativists when they are thinking about the weak field limit.

Two dimensional relativistic gravity

For the two dimensional gravity, with point masses, there is a nice description which can be understood immediately. Two dimensional point masses are parallel strings moving perpendicular to the direction of motion in $3d$ plus time, but these strings are like pencils of light, not stationary line-masses, they are relativistic along their direction of motion. You need to have a relativistic momentum density on the strings for them to reduce to the simple limit of 2+1 gravity.

In this limit, the strings are described by 2+1 gravity. The point masses in 2+1 gravity are described by cutting out a wedge from a two dimensional paper representing space-time, and gluing it back to form a cone. This description is exact--- this is what the space-time around a relativistic cosmic strings looks like. The space is called locally flat, because if you draw a least distance line it will be straight after unrolling the paper, so that the only curvature is that which can be seen from outside, not to a flat fellow living inside the paper. There is only intrinsic curvature at the tip of the cone, proportional to the deficit angle, the angular size of the wedge. This is is the mass of the string.

If you imagine a particle coming in from infinity, it travels in a straight line along the cone, but it comes out deflected in a certain way. This is easiest to see by taking two parallel lines coming in on opposite sides of the cone point--- they will intersect each other.

If you make a double-cone by cutting out two wedges, to make a slushie-shape after gluing. The paper is still locally flat, but if you draw two straight lines, the line passing between the cones will intersect the other lines. A collection of $n$ stationary cone points describes an equilibrium stationary configuration of $2d$ gravity.

If you set the cone points in motion, and add some points with negative curvature which evolve in a specific way (their curvature in the $3d$ sense is still zero) you get t'Hooft's description of 2+1 gravity, which is an active research subject today.