First, we need to create a spline that will define the shape of pieces. So go to Create panel -> Shapes -> select Splines from the drop down menu -> and finally select Line. Then just click on along the grid to create the line like on the picture (turn on Snap to Grid to get a good line).

Now we need to convert our line in to an Editable Spline. To do that, select the newly created line and right click on it. Select the option Convert To -> Convert to Editable Spline. Then, in the Interpolation settings check the Adaptive option so we can get smooth curves.

Move around the vertices so they match the next picture.

Now, select all of the vertices and then right click on one of them and change their type to Smooth.

Select the vertices that are marked on the picture and change their type to Bezier.

And change the next vertices to Corner.

Next, select one of the Bezier vertices and you can notice that two green handles appear. Select the Move tool from the toolbar and elongate them.

Note: You should track how much you move the handles, so you can elongate the handles for the same amount at every vertex. You have 3 field at the bottom of the screen that show you how much you have moved/rotated/scaled the select object/sub-object for every axes.

Do this for the other 3 vertices.

Next, apply the Extrude modifier to the Spline with the Amount of 15. This converted our 2d spline into the 3d object.

Add the Edit Poly to the modifier stack and select the top polygon.

Now, we need to bevel the selected polygon, twice. First time with the settings Height: 2.5, Outline Amount: -1.0; and the second time with Height: 2.5, Outline Amount: -1.5.

You should get something like this:

Apply the Edit Normals modifier to the object, select all normals and click on the the Unify button to unify them.

As you can notice, some artifacts appeared on our puzzle piece. To fix this, change the Select By option of the Edit Normals modifier to Face, click on the top face and then on the Break button.

Ok, we just created one piece of a puzzle. But because a whole puzzle is made up of different shaped pieces, we need to create couple more pieces. To do this, just return to the Line in the modifier stack and move the vertices's and if necessary, repeat the bevel and edit normal steps.

After you created couple of different pieces, arrange them in the position they would be in if the puzzle was completely assembled.

Then, create a material with a desired picture in the diffuse slot. Select all of the pieces and apply the material. Next, while all of the pieces are still selected, add the UVW Map modifier to them. Change the length/width in the modifier properties if needed.

Here is what i got when i rendered the puzzle:

Also, a good thing is that you can move the pieces around if you want, and their place in the complete puzzle (picture on the surface) will not change.