A simple, inexpensive, and clean way of securing a painting panel into a wooden frame is to use glazier points, which are available at nearly every hardware store. Glazier points come in several shapes, but those best suited for this use are small pieces of sheet metal, pointed at one end, with two flanges bent perpendicularly to the main body. These flanges provide purchase so that a screwdriver or putty knife can be used to drive the point into the wooden frame, or so that, with a pair of pliers, the point can later be removed more easily. Normally, glazier points are used to hold glass panes in window frames, but they are perfectly suited for this job as well.
Installing the glazier points is easy. After laying your picture frame face down on a soft surface to prevent any scratches to the finish, place your painting panel, also face down, inside the frame. Place a glazier's point flat side down on the back of the panel with the point facing the rabbet. Using a screwdriver tip positioned behind the vertical flanges, slide the point towards the inside of the frame. At this point, it will probably require a little finesse to insert the point: either work the point back and forward while applying pressure, or use a finishing hammer to lightly tap on the back of screwdriver to drive the point into the wood. To keep the frame from moving, you can place the side you are working on against a wall. If the point does not easily go into the frame, you may need to reposition the point left or right to find a softer section of wood. Repeat this process every few inches, or as is necessary according to the size of the frame.
Of course, using glazier points only works on frames which have a rabbet of greater depth than the thickness of the painting panel. Also, if the frame is plastic, or plastic coated wood, it will be much more difficult to drive the point into the frame, and glazier points might not be the best choice for your framing project.