We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:mrow></mml:math>. We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.