問題は「山折り谷折り」
この手の問題は関数型言語の方が向いているけれど、たまにはC#でも。
Program.cs
using System;
using System.Collections.Generic;
using System.Linq;
using Folding = System.Collections.Generic.Dictionary<char, string>;

class Program
{
    static void Main(string[] args)
    {
        // 省略
    }

    static void Test(string input, string expected)
    {
        string actual = Solve(input);
        Console.WriteLine(expected == actual ? "OK" : "NG");
    }

    static Folding front = new Folding { { 'L', "bVf" }, { 'J', "fVb" }, { 'Z', "fmbVf" }, { 'U', "bVfVb" }, { 'S', "fVbmf" } };
    static Folding back = new Folding { { 'L', "bmf" }, { 'J', "fmb" }, { 'Z', "bmfVb" }, { 'U', "fmbmf" }, { 'S', "bVfmb" } };

    static string Solve(string input)
    {
        var folds = input
            .Aggregate("f".AsEnumerable(),
                       (xs, y) => xs.SelectMany(x => Fold(x, y)))
            .Where(x => x == 'm' || x == 'V');
        return new string(folds.ToArray());
    }

    static IEnumerable<char> Fold(char current, char input)
    {
        switch (current)
        {
            case 'f':
                return front[input];
            case 'b':
                return back[input];
            case 'm':
            case 'V':
                return new[] { current };
            default:
                return "";
        }
    }
}