Unless you're using special .NET classes or native libraries it should be possible and easy.
Since the .NET compiled executables are built on bytecode, they aren't linked to a specific platform and Mono has been designed with this in mind.
Supposing you've just tested an application named "WindowsApplication", try to follow these steps:
Check under your Visual Studio projects folder, try to locate your WindowsApplication.exe.
Check "C:\Users\YourName\Documents\Visual Studio\Projects\WindowsApplication\WindowsApplication\bin\Debug
(or Release)
After installing Mono on your Ubuntu system check if the mono command is available in your folder (for this test just use your home dir):
username@locahost ~
$ mono --version
Mono JIT compiler version 5.0.1 (Visual Studio built mono)
Copyright (C) 2002-2014 Novell, Inc, Xamarin Inc and Contributors. www.mono-project.com
TLS: normal
SIGSEGV: normal
(more output)
Copy your program executable from Windows into the Ubuntu folder, let's suppose you copied to the directory where you just tested mono.
Launch it by typing:
mono ./WindowsApplication1.exe
(if it's in your current directory)
or
mono /<another_path_to_your_executable>/WindowsAppliction.exe
In case it didn't run you can download the "Mono Migration Analyzer" (MoMA), which is a tool specifically designed to identify unresolved dependencies and help you solve the problem.