How to get raggedright in tabular*?

Add \arraybackslash after \raggedright:

\documentclass{article}
\usepackage{array}
\newcommand\fox{The quick brown fox jumps.}

\begin{document}
\begin{tabular*}{5in}{%
  >{\raggedright\arraybackslash}p{1.5in}%
  >{\raggedright\arraybackslash}p{1.5in}%
  >{\raggedright\arraybackslash}p{1.5in}}% does work in 3rd column :)
  \fox & \fox & \fox \\
  \fox & \fox & \fox
\end{tabular*}
\end{document}

You can shorten the code a bit by repeating the same column 3 times:

\documentclass{article}
\usepackage{array}
\newcommand\fox{The quick brown fox jumps.}

\begin{document}
\begin{tabular*}{5in}{%
    *{3}{>{\raggedright\arraybackslash}p{1.5in}}}%
  \fox & \fox & \fox \\
  \fox & \fox & \fox
\end{tabular*}
\end{document}

if you use \tabularnewline, than you not need \arraybackslash for restoring meaning of \\:

\documentclass{article}
\usepackage{array}
\newcommand\fox{The quick brown fox jumps.}

\begin{document}
\begin{tabular}{*{3}{>{\raggedright}p{1.5in}}}%
  \fox & \fox & \fox    \tabularnewline
  \fox & \fox & \fox
\end{tabular}
\end{document}

off-topic: if you define width of all columns in table, than you not need to define table width with use tabular* environment, since it is determined by column widths. exception is, if you like to add @{\extracolsep{\fill}} by which you spread columns over prescribed table width.

You can also put \raggedright after p. I've just used it for the last column but it would be OK with the other columns. For more details see the descriptions mentioned by @DavidCarlisle in Auto Detect Last Paragraph and Pass to Macro.

\documentclass{article}
\usepackage{array}
\usepackage{lipsum}
\newcommand\fox{The quick brown fox jumps.}

\begin{document}
\begin{tabular*}{5in}{%
  >{\raggedright}p{1.5in}%
  >{\raggedright}p{1.5in}%
  p{1.5in}<{\raggedright}}
%  >{\raggedright}p{1.5in}} % doesn't work in 3rd column
  \lipsum[1] & \lipsum[1] & \lipsum[1] \\
  \fox & \fox & \fox
\end{tabular*}
\end{document}