Large scale albedo changes
Albedo is the reflectivity of solar radiation off the surface of a planet or moon or something. It represents how much of the solar radiation is sent back into space as opposed to how much is retained.
The planetary albedo of the Earth is about 0.367. However, this varies greatly from place to place. Cumulus clouds and fresh snow can have an albedo of 0.7 or more; that is, 70% of incident radiation is reflected right back into space. On the other hand the ocean's albedo is about 0.08, which means that 92% of the solar radiation striking the ocean stays with it.
There are certain things that humans can do the change the albedo. Most plausibly for ancient man, a forest might have an albedo of 0.1 while dry bare soil might be 0.3. Thus, if Man could change the environment from one to the other over a large enough area, he could change the Earth's albedo.
How much area is reasonable for Man to change?
There were some large scale deforestation incidents in the ancient world. Possibly the biggest was the reduction of the Gangetic Plains. 10,000 years ago the area from Delhi to Bengal was covered in moist deciduous tropical forest; an area of around 500,000 km$^2$. This was ultimately replaced by cropland over the period from roughly 1000 BC to 1000 AD. Crops and irrigated soil have an albedo not too much higher than a forest, so the climate change impact wasn't much. But that Ganges plain is actually pretty dry; had humans not spent so much time irrigating it could have ended up a barren semi-desert like the African Sahel.
This 500,000 km$^2$ represents 0.1% of the Earth's surface. If .1% of the Earth's surface increased in albedo by 20 percentage points, the overall Earth's albedo would have changed from 0.367 to about 0.3677; not much.
But what is super self destructive humans did this to similar decidous tropical forests around the world? There are 2.7 million km$^2$ of miombo woodland in southern Africa; about 0.7 million km$^2$ more in India's northern Deccan Plateau; 0.4 million in Indochina; 0.3 million on the Pacific coast of Central America; and 0.9 million in the Dry Chaco of Argentina and Bolivia. This is about 5.5 million km$^2$ of forest total, or up to 1.1% of the Earth's surface. The albedo effect of turning all this forest into barrens would be from 0.367 to 0.375.
What does that albedo change do?
Albedo can be used along with stellar luminosity and orbital distance to calculate planetary effective temperature using
$$T_{eff} = \sqrt[4]{\frac{1}{4}\frac{L(1-a)}{4\pi\epsilon\sigma R^2}}$$ where $\sigma$ is the Stefan-Boltzman constant ($3.67\times10^{-8} \text{W m}^{-2}\text{K}^{-4}$); $\epsilon$ is planetary emissivity (0.96 for Earth); $R$ is distance from the sun ($1.50\times10^{11} \text{ m}$); $L$ is the luminosity of the sun ($3.83\times10^{26} \text{ W}$); and $a$ is albedo.
For Earth's albedo of 0.367, $T_{eff} = 276.5 \text{ K}$. This is a bit low; planetary average temp is more like 285 K, and the difference is mostly due to the greenhouse effect. But it is close enough to demonstrate the changes. When albedo raises to 0.375, $T_{eff} = 275.6 \text{ K}$. This is about 1 degree of cooling from albedo changes, or about the same magnitude as the heating that we have currently caused due to climate change.
Conclusion
The ancients had it in their power to change the climate the same amount that we have changed it today. However, this will require them to cut down some 5 million km$^2$ of forest; an area about twice the size of Argentina. And not just cut it down, leave it fallow so its turns into a scrubby desert with lots of exposed dirt.
Now, humanity certainly has the capacity to do this, even with stone age technology. All these forests I listed have dry seasons of more than 6 months. It wouldn't be too much work to set everything on fire at the end of the dry season. As a double bonus, all these forests have heavy monsoonal rains during the wet season, so if you timed it just right, you could get the monsoon to wash all the ashes away, leaving parched, unfertile soil that would take centuries to recover.
This seems a bit extreme, but given the sorts of things we have and are doing to the Earth, maybe we should consider ourselves lucky this didn't happen already.