Two solutions involving XeLaTeX
Polyglossia
With Polyglossia the hyphenations rules are loaded correctly for all the declared languages.
\documentclass{article}
\usepackage{polyglossia}
\setmainlanguage{english}
\setotherlanguage[variant=mono]{greek}
% Fonts
\setmainfont[Mapping=tex-text]{Linux Libertine O}
\begin{document}
\section{Τίτλος Κεφαλαίου}
English text. % Main language doesn't need declaration.
\begin{greek} % Secondary languages need to be declared.
Ελληνικό κείμενο.
\end{greek}
\end{document}
You could probably define a shortcut for the greek environment, like
\newcommand{\gr}[1]{\begin{greek}#1\end{greek}}
...
\gr{Και άλλο ελληνικό κείμενο}
...
xgreek
With xgreek you can use upright letters for both languages out of the box. Keep in mind that this package is mainly used for the typesetting of greek texts. If the main language of your documents is english, better stick with polyglossia.
In your case, the main problems with xgreek is that in order to have hyphenation in english, you must explicitly declare the text as english one and that chapters, sections, captions etc would be translated in greek.
\documentclass{article}
\usepackage{fontspec}
\usepackage{xgreek}
\setmainfont{Linux Libertine O}
% This command allows the use of both english and greek text with correct hyphenation.
% English text must be placed in braces. Example:
% Ελληνικό κείμενο. \en{English text.} Ελληνικό κείμενο.
\newcommand{\en}[1]{\setlanguage{american}#1\setlanguage{monogreek}}
\begin{document}
\en{I want to use the upright Greek letters in the actual font being used and I want the hyphenation rules to work correctly!}
Οι ελληνικοί κανόνες συλλαβισμού δουλεύουν χωρίς προβλήματα.
\end{document}