Reaction speed with solids
Reaction rates are in most cases influenced by concentration for liquids/dissolved solids and surface area for solids. 

Surface area

The surface area of the calcium carbonate is one factor in the rate of the reaction. The more surface area, the faster the reaction. In this case, certainly, the five $\pu{1kg}$ cubes will react much faster. The reason large cubes are used to demonstrate this instead of powder is because the surface area is more or less constant throughout the reaction.

The rate law
$$\text {Rate} = k[\text A]^\text m[\text B]^\text n $$ where $\text A$ and $\text {B}$ are the reactant concentration, $k$ is the rate constant (does not change in the same reaction), and $m$ and $n$ are the rate orders (which also does not change in a reaction)

This means that unless m and n are 0 making it a zero order reaction which is independent of concentration, in most cases concentration does have an effect on rate.

In this case, with $k$, m and n constant and either  m or n non-zero, and increase in $[\text A] \text{ or } [\text B]$ would increase the rate.

If $[\text A]$ is calcium carbonate and $[\text B]$ is hydrochloric acid, it shows that most likely an increase in concentration of $\text B$ will increase rate. However, calcium carbonate is a solid and has no concentration. We can imagine $[\text A]$ as the surface area of $\ce{CaCO3}$ instead, and m as 1.

We could also imagine the $\ce{CaCO3}$ dissolved in a solvent, like sugar in water. Now, the reaction rate will be dependent on the amount of 
$\ce{CaCO3}$
(Note: the rate orders can only be determined experimentally)