Running a script recursively in subdirectories
Check yourself before you wreck yourself...

shellcheck.net is an awesome website that checks your Bash syntax.
Among other things, it will tell you to:


Double-quote your path variables
Use $(...) instead of the legacy `...` for sub-shells


Avoid cd inside scripts

cd inside scripts is troublesome.
In case something goes wrong,
you might find yourself in unexpected places, and wreak havoc.

But if you must, like in your example,
there's a better way than saving the previous directory in PREV_DIR and then cd "$PREV_DIR" to go back, using a (...) sub-shell environment:

(
cd "$f" || break
RESULT=$(eval "$RUN_SCRIPT")
local RESULT_CODE=$?
if [ $RESULT_CODE -eq 0 ]; then
    echo "$f"
    echo "$RESULT"
    echo
else
    recurseCheck "$f"
fi
)


Notice that there is only one cd in, no cd out.
This is because the cwd changes only affect the environment inside the (...).

Also notice the cd "$f" || break: if the cd command fails for some reason,
you probably don't want to execute the rest of the block.
You might go as far as exit at that point (shellcheck recommends it too) and investigate the troublesome directory.

Minor things

You had a pointless ; in done;, and instead of echo "" you can write simply echo.

Input validation

What if you run this script without arguments?
eval "" exits with success, so it will simply print all directories.
If that's ok, then everything's fine.
If not, then you might want to add some input validation, for example:

if [ ! "$RUN_SCRIPT" ]; then
    echo usage: $0 'some script'
    exit 1
fi


Suggested implementation

Putting it together (coming to you in a Pull Request very soon):

#!/bin/bash

RUN_SCRIPT=$1

recurseCheck() {
    local f
    for f in $1/* ; do
        if [ -d "$f" ]; then
            (
            cd "$f" || break
            RESULT=$(eval "$RUN_SCRIPT")
            local RESULT_CODE=$?
            if [ $RESULT_CODE -eq 0 ]; then
                echo "$f"
                echo "$RESULT"
                echo
            else
                recurseCheck "$f"
            fi
            )
        fi
    done
}

START_DIR=$(pwd)
recurseCheck "$START_DIR"