3600
【Solution】The edge length of the cube should be the least common multiple of $5, 4, 3$. The least common multiple of $5, 4, 3$ is 60; therefore, the minimum number of such rectangular wooden blocks needed is:
$$
\begin{aligned}
& (60 \times 60 \times 60) \div(5 \times 4 \times 3), \\
= & 216000 \div 60, \\
= & 3600 \text { (blocks) }
\end{aligned}
$$

Answer: At least 3600 such rectangular wooden blocks are needed.
So the answer is: 3600.
19. (6 points) Use rectangular wooden blocks that are 5 cm long, 4 cm wide, and 3 cm high to stack into a cube. At least how many of these rectangular wooden blocks are needed.

To maintain the original format and line breaks, the translation is as follows:

19. (6 points) Use rectangular wooden blocks that are 5 cm long, 4 cm wide, and 3 cm high to stack into a cube. At least how many of these rectangular wooden blocks are needed.