It seems as if the rectangle background becomes black when printing and by using black hatched lines, the whole rectangle becomes black. I avoided this problem by using grey lines and define my own pattern as indicated by eiterorm.
Example as follows:
\documentclass[10pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
%\usetikzlibrary{backgrounds}
\usepackage{tikz}
\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 west 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}
}
\begin{document}
\begin{center}
%
\begin{tikzpicture}
\draw[pattern=custom north west lines] (0,0) rectangle (3,4);
\end{tikzpicture}%
%
\begin{tikzpicture}
\draw[pattern=custom north west lines,hatchspread=5pt,hatchthickness=2pt,hatchcolor=gray!20,] (0,0) rectangle (3,4);
\end{tikzpicture}%
%
\begin{tikzpicture}
\draw[pattern=custom north west lines,hatchspread=8pt,hatchthickness=5pt, hatchcolor=gray!20] (0,0) rectangle (3,4);
\end{tikzpicture}%
%
\begin{tikzpicture}
\draw[pattern=custom north west lines,hatchspread=10pt,hatchthickness=3pt, hatchcolor=gray!20] (0,0) rectangle (3,4);
\end{tikzpicture}%
\end{center}
\end{document}
There might be a more elegant solution by changing the actual background color though.