Below I give you some possibilities using both mdframed and tcolorbox:
The tcolorbox option:
\documentclass{article}
\usepackage[many]{tcolorbox}
\usetikzlibrary{patterns}
% defining the new dimensions and parameters
\newlength{\hatchspread}
\newlength{\hatchthickness}
\newlength{\hatchshift}
\newcommand{\hatchcolor}{}
% declaring the keys in tikz
\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
hatchthickness/.code={\setlength{\hatchthickness}{#1}},
hatchshift/.code={\setlength{\hatchshift}{#1}},% must be >= 0
hatchcolor/.code={\renewcommand{\hatchcolor}{#1}}}
% setting the default values
\tikzset{hatchspread=3pt,
hatchthickness=0.4pt,
hatchshift=0pt,% must be >= 0
hatchcolor=black}
% declaring the pattern
\pgfdeclarepatternformonly[\hatchspread,\hatchthickness,\hatchshift,\hatchcolor]% variables
{custom north east lines}% name
{\pgfqpoint{\dimexpr-2\hatchthickness}{\dimexpr-2\hatchthickness}}% lower left corner
{\pgfqpoint{\dimexpr\hatchspread+2\hatchthickness}{\dimexpr\hatchspread+2\hatchthickness}}% upper right corner
{\pgfqpoint{\dimexpr\hatchspread}{\dimexpr\hatchspread}}% tile size
{% shape description
\pgfsetlinewidth{\hatchthickness}
\pgfpathmoveto{\pgfqpoint{0pt}{\dimexpr\hatchspread+\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr\hatchspread+0.15pt+\hatchshift}{-0.15pt}}
\ifdim \hatchshift > 0pt
\pgfpathmoveto{\pgfqpoint{0pt}{\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr0.15pt+\hatchshift}{-0.15pt}}
\fi
\pgfsetstrokecolor{\hatchcolor}
% \pgfsetdash{{1pt}{1pt}}{0pt}% dashing cannot work correctly in all situation this way
\pgfusepath{stroke}
}
\newtcolorbox{stripbox}{
enhanced,
frame code={},
interior code={
\path[
draw=green!80!black,
rounded corners,
pattern=custom north east lines,
hatchspread=12pt,
hatchthickness=4pt,
hatchcolor=gray!20
]
(interior.south east) rectangle (interior.north west);
}
}
\begin{document}
\begin{stripbox}
test text
\end{stripbox}
\end{document}
The output:

The mdframed option:
\documentclass{article}
\usepackage[framemethod=tikz]{mdframed}
\usetikzlibrary{patterns,backgrounds}
% defining the new dimensions and parameters
\newlength{\hatchspread}
\newlength{\hatchthickness}
\newlength{\hatchshift}
\newcommand{\hatchcolor}{}
% declaring the keys in tikz
\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
hatchthickness/.code={\setlength{\hatchthickness}{#1}},
hatchshift/.code={\setlength{\hatchshift}{#1}},% must be >= 0
hatchcolor/.code={\renewcommand{\hatchcolor}{#1}}}
% setting the default values
\tikzset{hatchspread=3pt,
hatchthickness=0.4pt,
hatchshift=0pt,% must be >= 0
hatchcolor=black}
% declaring the pattern
\pgfdeclarepatternformonly[\hatchspread,\hatchthickness,\hatchshift,\hatchcolor]% variables
{custom north east lines}% name
{\pgfqpoint{\dimexpr-2\hatchthickness}{\dimexpr-2\hatchthickness}}% lower left corner
{\pgfqpoint{\dimexpr\hatchspread+2\hatchthickness}{\dimexpr\hatchspread+2\hatchthickness}}% upper right corner
{\pgfqpoint{\dimexpr\hatchspread}{\dimexpr\hatchspread}}% tile size
{% shape description
\pgfsetlinewidth{\hatchthickness}
\pgfpathmoveto{\pgfqpoint{0pt}{\dimexpr\hatchspread+\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr\hatchspread+0.15pt+\hatchshift}{-0.15pt}}
\ifdim \hatchshift > 0pt
\pgfpathmoveto{\pgfqpoint{0pt}{\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr0.15pt+\hatchshift}{-0.15pt}}
\fi
\pgfsetstrokecolor{\hatchcolor}
% \pgfsetdash{{1pt}{1pt}}{0pt}% dashing cannot work correctly in all situation this way
\pgfusepath{stroke}
}
\tikzset{
}
\newmdenv[
nobreak,
roundcorner=6pt,
linewidth=0pt,
apptotikzsetting={
\tikzset{mdfbackground/.append style=
{
draw=green!80!black,
rounded corners,
pattern=custom north east lines,
hatchspread=12pt,
hatchthickness=4pt,
hatchcolor=gray!20
}
}
}
]{stripbox}
\begin{document}
\begin{stripbox}
test text
\end{stripbox}
\end{document}
The output:

Remarks
The customization for the pattern was done using Philippe Goutet's code in his answer to Custom and built in TikZ fill patterns.
Since the code in the question includes the nobreak option I assumed in my answer that the environment is not required to admit page breaks; if this is not the case, the modifications required are not too big and can be easily made.
\strutto the last line should solve that. – barbara beeton Jun 11 '15 at 17:48:)Sorry for the unequal spacing, I wrote a quick and dirty example! You are absolutely right,\strutdoes solve it; I can also add ainnerbottommarginoption to mymdframedenvironment and make things display as expected. – Paulo Cereda Jun 11 '15 at 17:54:)– Paulo Cereda Jun 11 '15 at 19:09arara v4manual? :) – cmhughes Jun 12 '15 at 09:28:)I'm planning something along these lines, but a huge block with a striped background doesn't look so pleasant to the eye (it works with smaller chunks), so I have to use it carefully.:)By the way, we are making some huge progress on the tool, it's more stable than ever and with very cool features, I am sure you will love it.:)– Paulo Cereda Jun 12 '15 at 10:25