Do you still use Java 1.3.1?
Ten this issue can appear because of some known bugs in the implementation of DirectX in some display drivers (notably on some notebooks).
Updating to DirectX 9.0c may solve the problem.
But Java 1.4.2 or Java 1.5 contain now some workarounds for such display driver bugs that do not properly sequence the various graphic primitives (in your case, it seems that your display driver delays the rectangle-fill primitive by hardware, but performs immediately other graphic software primitives, and the display driver has conflicts when sequencing hardware and non-hardware graphic primitives, notably if they come from separate graphic API, such as GDI and DirectDraw).

So the solution for your case is:
1.a) update your graphic driver to get the best compatibility with DirectX 8 at least (9 would be best)
1.b) download the DirectX 9.0c from Windows Update (apparently you use Windows XP)
2) update to Java 1.4.2 or even 1.5, visit www.java.sun
3) open the Java control panel, in the "Java" tab, to make sure that the last version of the Java JRE is selected; if there are multiple entries, make sure to select the JRE which you installed the last.
You should uninstall the JRE 1.3.1 if it's still there. If there's no more uninstall row in the "Remove Application" control panel, then you can simply delete the "jre" directory where it is stored (in the old International versions of Limewire, the bundled JRE was created in a "jre" subdirectory of LimeWire, but it's best to install Java independantly before installing LimeWire)
4) Make sure that you can successfully run the online demos on the Java donwload web site (the "showcase" section): try some free games, interactive maps. Make sure that the demonstration JNLP applets run properly (notably the Swing demo, in the scrollable tables or scrollable images, which seems to be the most critical case where DirectX driver bugs are visible).

If all these do not work properly, you may try by disabling some hardware acceleration in the Display control panel. Try also by disabling the mouse cursor shadow (there are known problems with this mouse feature with some hardwares, because of conflicts with the hardware mouse cursor acceleration and the sofware shadowing of the cursor which requires alpha-transparency support sometimes emulated by software).

5) reinstall LimeWire with this new Java setting...